Controllability is an important property of a control system and plays a crucial role in many regulation problems, such as the stabilization of unstable systems using feedback, tracking problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state.
Controllability and observability are dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed).
Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with.
The following are examples of variants of notions of controllability that have been introduced in the systems and control literature:
State controllability: the ability to steer the system between states Strong controllability: the ability to steer between states over any specified time window Collective controllability: the ability to simultaneously steer a collection of dynamical systems Trajectory controllability: the ability to steer along a predefined trajectory rather than just to a desired final state Output controllability: the ability to steer to specified values of the output Controllability in the behavioural framework: a compatibility condition between past and future input and output trajectories
State controllability
The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.
(or simply if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.
That is, we can informally define controllability as follows: If for any initial state and any final state there exists an input sequence to transfer the system state from to in a finite time interval, then the system modeled by the state-space representation is controllable. For the simplest example of a continuous, linear time-invariant (LTI) system, the row dimension of the state space expression determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of , then the system cannot achieve controllability. It may be necessary to modify and to better approximate the underlying differential relationships it estimates to achieve controllability.
Controllability does not mean that a reached state can be maintained, merely that any state can be reached.
Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within a finite time interval. When the time interval can also be specified, the dynamical system is often referred to as being strongly controllable.
Continuous linear systems
Consider the continuous linear system A linear time-invariant system behaves the same but with the coefficients being constant in time.
There exists a control from state at time to state at time if and only if is in the column space of
where is the state-transition matrix, and is the Controllability Gramian.
In fact, if is a solution to then a control given by would make the desired transfer.
Note that the matrix defined as above has the following properties:
Rank condition for controllability The Controllability Gramian involves integration of the state-transition matrix of a system. A simpler condition for controllability is a rank condition analogous to the Kalman rank condition for time-invariant systems.
Consider a continuous-time linear system smoothly varying in an interval :
The state-transition matrix is also smooth. Introduce the matrix-valued function and define
Consider the matrix of matrix-valued functions obtained by listing all the columns of the matrix (, for ):
If there exists a and a nonnegative integer such that , then is controllable.Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
If is also analytically varying in an interval , then is controllable on every nontrivial subinterval of if and only if there exists a and a nonnegative integer such that .
The above methods can still be complex to check, since it involves the computation of the state-transition matrix . Another equivalent condition is defined as follow. Let , and for each , define
In this case, each is obtained directly from the data . The system is controllable if there exists a and a nonnegative integer such that .
Example
Consider a system varying analytically in and matrices
Then
and since this matrix has rank 3, the system is controllable on every nontrivial interval of
Continuous linear time-invariant (LTI) systems
Consider the continuous linear time-invariant system
where is the state vector; is the output vector; is the input (or control) vector; is the state matrix; is the input matrix; is the output matrix; and is the feedthrough (or feedforward) matrix.
The controllability matrix is given by
The system is controllable if the controllability matrix has full row rank (i.e. .
Discrete linear time-invariant (LTI) systems
For a discrete-time linear state-space system (i.e. time variable the state equation is
where is an matrix and is a matrix (i.e. is inputs collected in a vector). The test for controllability is that the matrix
has full row rank (i.e., ). That is, if the system is controllable, will have columns that are linearly independent; if columns of are linearly independent, each of the states is reachable by giving the system proper inputs through the variable .
Derivation
Given the state at an initial time, arbitrarily denoted as , the state equation gives , then , and so on with repeated back-substitutions of the state variable, eventually yielding
or equivalently
Imposing any desired value of the state vector on the left side, this can always be solved for the stacked vector of control vectors if and only if the matrix of matrices at the beginning of the right side has full row rank.
Example For example, consider the case when and (i.e. only one control input). Thus, and are vectors. If has rank 2 (full rank), and so and are linearly independent and span the entire plane. If the rank is 1, then and are collinear and do not span the plane.
Assume that the initial state is zero.
At time :
At time :
At time all of the reachable states are on the line formed by the vector .
At time all of the reachable states are linear combinations of and . If the system is controllable then these two vectors can span the entire plane and can be done so for time .
The assumption made that the initial state is zero is merely for convenience. Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).
This example holds for all positive , but the case of is easier to visualize.
Analogy for example of = 2 Consider an analogy to the previous example system. You are sitting in your car on an infinite, flat plane and facing north. The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line.
If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north). The lack of steering case would be analogous to when the rank of is 1 (the two distances you drove are on the same line).
Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of is 2.
If you change this example to then the analogy would be flying in space to reach any position in 3D space (ignoring the orientation of the aircraft).
You are allowed to: fly in a straight line turn left or right by any amount (Yaw) direct the plane upwards or downwards by any amount (Pitch) Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.
Nonlinear systems Nonlinear systems in the control-affine form
are locally accessible about if the accessibility distribution spans space, when equals the dimension of and R is given by:Isidori, Alberto (1989). Nonlinear Control Systems, p. 92–3. Springer-Verlag, London. .
Here, is the repeated Lie bracket operation defined by
The controllability matrix for linear systems in the previous section can in fact be derived from this equation.
Controllability via state feedback When control authority on a linear dynamical system is exerted through a choice of a time-varying feedback gain matrix , the system
is nonlinear, in that products of control parameters and states are present. The accessibility distribution is, as before,
It is clear that for the system to be controllable, it is necessary that has full column rank. It turns out that this condition is also sufficient. However, the (optimal) control strategy explained earlier needs to be modified so that the trajectory when applying an optimal input to steer the system between the specified states, does not pass through the origin, else the regulating input cannot be written in feedback form . The controllability, as well as the strong controllability of this bilinear system was proven in .
Collective controllability: feedback control of state transition
Collective controllability represents the ability to steer linear dynamical systems that obey identical dynamics
where equals the dimension of , between specified starting and ending configurations by way of a common state feedback gain matrix , and thereby, each instantiating a control input
for , respectively.
The accessibility distribution having full column rank is trivially a necessary condition. It is also sufficient, and in fact, the collective is strongly controllable, in that it can be steered from an initial configuration
to any specified terminal configuration
provided , over any specified time interval through a choice of a common time-varying feedback gain matrix provided has full column rank.
Null controllability If a discrete control system is null-controllable, it means that there exists a controllable so that for some initial state . In other words, it is equivalent to the condition that there exists a matrix such that is nilpotent.
This can be easily shown by controllable-uncontrollable decomposition.
Output controllability
is the related notion for the output of the system (denoted in the previous equations); the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular: